There's no infinitive with sagen: "Ich sagte ihr zu kommen" is not possible. This type of construction is possible with a large number of other verbs, e.g. befehlen, bitten, ersuchen.
Without soll, the sentence doesn't make sense as it wouldn't be clear that she is to do something, i.e. come. Solle is first subjunctive because it's reported speech, this is for 55-year old teachers like myself. Möge makes the "order" for her to come more polite.

It's not that difficult: "Zu" is - in that context - like the English "to", it just cannot be used in all the same cases. And "dass" is like the English "that", or the Spanish "que".
I think in Spanish most of the cases are formed with the subjunctive ("Le dije que viniera", "Le pedí que viniera", no?). In English, those can both be formed with "to" + infinitive: "I told her to come", "I asked her to come".
In German, the "zu" + Infinitiv form works in fewer cases than in English. I'm afraid you'll simply have to learn the verbs it works with. But it never works with sagen: "Ich sagte ihr, dass sie kommen soll" or "Ich sagte ihr, sie soll kommen". (literally: "Le dije que debe venir"; "I told her that she should come") The form with "sollen" is often used when in English the form "tell somebody" + "to" + infinitive is used.

That's grammatically correct, but it means something else. You're not telling her to come; she is coming and that's a fact. (English: I told her (that) she's coming/she will come. Spanish: Le dije que (ella) viene/venía/vino.)

I'm afraid in the German version of your sentence there's no way around the "sollen".

That means "I told her that she's comming" and that, expect for a single case I can think of, doesn't make any sense. Why would you tell her that? If she's coming, she'll know that herself.

There is, however, a case were one might say that: If she always denied that she would come but you have told her that eventually she would be coming, and in the end she really showed up, then you might say: "Ich habe ihr (immer) gesagt, dass sie kommt (käme/kommen würde)."